Can companies use reverse stock splits to liquidate my shares at unfavourable prices?

The company doesn't do this type of maneuver. The board of directors of the company does this type of maneuver. Depending on the articles of incorporation the board may need stockholder approval. If they do require a vote that could be done at an annual meeting or at a special shareholder meeting.

The goal when they are getting rid of small investors is to end up with a low enough number so that they can be considered private. That reduces the expense of the required paperwork.

Yes this maneuver can be expensive, but the board determined that this path is the best path forward.

This is an example of one US company that did this:

To deregister its shares of common stock, Gander Mountain will reduce its number of beneficial shareholders to fewer than 300 through reverse and forward stock splits. Shareholders will receive a cash payment of $5.15 for each share.

The funding for the cash payment for the fractional shares will be provided by the company’s two largest shareholders, Gratco LLC and Holiday Stationstores. Greene Holcomb & Fisher determined that the cash-out price of fractional shares is fair from a financial point of view to those shareholders.

Nothing of the sort. When they do that sort of thing, they put notice out to the market first (and if you’re holding a single stock outside of an index fund, it’s basically your job to remain attuned to such news). Their notice will tell you what you need to do if you want to keep your 700 shares invested in the company.

At that point, you will have the option to do as you please. For instance, you could

• sell the 700 shares and quit the stock, or let this happen passively
• Buy 300 more shares to have an even 1000, finishing the reverse split with 1 share. Since you believe the stock is wildly undervalued, this is a perfectly reasonable thing to do, yes?
• Let them cashier you out of the 700 shares, and then turn around and immediately buy 1 of the new shares. As long as you do it quickly, its value isn’t likely to change a whole lot.

The purpose of the reverse split isn’t to shake off trivial customers, it’s to rescope the stock value to what the markets require.

Let’s say you’re Oceanic Air Lines, one of the great historic airlines, like Pan Am, with $50 billion a year of business. You have a massive commercial network with 1200 gates at 273 airports around the world, and led the development of facilities at many of them. As a result, you have very healthy (well, $250 million) FBO, fueling and catering sideline businesses. The businesses are properly segmented in independent LLCs. Recent events drove the airline LLC bankrupt. So now you are a successful but rather small company compared to what you once were. Your stock price reflects this. NYSE is happy to have a $250M company, but does not like penny stocks. As a result, you’ll be kicked off the exchange unless you get your 11-cent-a-share stock price to at least $10/share. What do you do?

Yes, this is possible under the right circumstances. For example: In 2018, a company called "IEG Holdings Corporation", doing business as "Mr. Amazing Loans" made what's called a "tender offer" to owners of LendingClub stock. (I received this offer at the time.) You can read the text of the offer here: Form S-4 IEG Holdings Corp.

The offer handwaved some numbers on the first page, alleging that the "AMOUNT YOU MAY POTENTIALLY GAIN" was 19 cents per share. Reading the actual conditions of the offer made it obvious that the reality was very different, and anybody tendering to the offer would most likely be doing so at a substantial loss. LendingClub put out a press release detailing some of this, and pointing to similar past offers by the same offeror which had negative consequences for people who accepted them: LendingClub Cautions Investors Against Potential Unsolicited Exchange Offer from IEG Holdings.

The offering document, in particular, discloses the following:

IEG Holdings has executed and initiated reverse stock splits that have resulted in a reduction in the number of stockholders of IEG Holdings. Lending Club stockholders who [accept] may be involuntarily cashed out in the future at a price per share that is lower than the price at which the IEG Holdings shares are valued in the Offer. In April 2016 and October 2016, we effected reverse and forward stock splits. In each case, stockholders who would otherwise have received fractional shares received cash in exchange for shares of IEG Holdings. These stockholders were involuntarily cashed out ... If we decide to execute a reverse stock split following the Offer, ... Lending Club stockholders who tender their shares in the Offer ... could be required to accept cash in exchange for their shares of IEG Holdings. If, for example, the stock price of IEG Holdings common stock is at that point below the price at which the IEG Holdings common stock is valued in this Offer, such stockholders could receive consideration for their IEG Holdings common stock that is less than the value of their tendered Lending Club shares at the time of the Offer.

IEG Holdings appears to be a public company, but it does not trade on any stock exchange — it's a "penny stock", trading over-the-counter, so the market in it is not very liquid; and it appears to be effectively under the control of a single person. Anybody who accepted this superficially-favorable tender offer would ultimately have not much option but to be — in the offer's own words — "involuntarily cashed out" by a reverse split subsequently.

Although I am not a lawyer, I would be shocked if there was not something in all this which the SEC could find fault with. However, as the offer was ultimately withdrawn, it did not get litigated. The company was apparently successful in a similar past offer (as described in LendingClub's press release warning).

I agree with other answers that this kind of monkey business is unusual, and that most reverse splits are unlikely to be so blatantly against the interests of the stockholders as the one described here. (And frankly it's not the worst part of what I have described here, by a longshot — accepting the tender offer is the mistake here, the reverse split is just a technical detail.) But it's still something to watch out for.