Alcohols like rum or vodka aren’t going to freeze on a normal day. Pure ethanol alcohol freezes at around -114°C (-173.2°F). But beer isn’t pure ethanol. It's made up mostly of water; and water freezes at 0°C or 32°F.
The temperature beer turns to slush or to ice depends on the alcohol content. Typically, 4-6% ABV beer is going to freeze at -2°C (28°F), but different styles of beer contain varying levels of alcohol.
If you want to chill your beer quickly without the wait—and danger of freezing—take a wet paper towel. Wrap the towel around the bottle then stick it in the freezer. You should have chilled beer in 10 minutes.