Can stocks trade at negative prices?

With an unlimited liability company, yes, in theory, its stock can trade at negative value that will still be capped by personal bankruptcy...in other words, if you own a portion of an unlimited liability company that is worth -$1 million, and your net worth is $250k, you'll just have to file personal bankruptcy and pay what can be recovered from your assets. The obvious difference with limited liability companies is that this firewall exists at the company level (rather than the individual) i.e. company files bankruptcy and your stockholdings get devalued to zero if liabilities exceed assets.

Having said all that, unlimited liability companies are quite rare if any on public exchanges. I believe most exchanges preclude them from being listed, but I can't find anything to support that belief. The last notable unlimited liability company I could find was American Express which converted to limited liability in 1965. This article states that it was the last publicly traded unlimited liability company in the United States.

A lot of the unlimited liability companies you hear of today are sole proprietors or partnerships. When you see a publicly traded corporate name/brand associated with the unlimited liability designation, that specific entity is most likely a subsidiary of some sort, probably created in that form for tax purposes, and surrounded by legal firewalls that prevent shareholder exposure to unlimited liability.