If I loan securities to my broker, can this turn long-term capital gains/losses into short-term capital gains/losses?

Loaning your securities has nothing to do with capital gains. You buy the stock on date "X" and sell it on date "Y". The holding period is the time between "X" and "Y" and the capital gain/loss is determined by your purchase and sale price.

What can change is how your dividends are taxed if you receive any. The shorter who borrows your stock must pay you the amount of the dividend if he is short on the ex-dividend date. This is payment is payment-in-lieu and it will be taxed as ordinary income and you have lost qualified dividend status.