My dog's eye is swollen after a walk. Should I take him to the vet?

Poor Alpha! The most common reasons for swollen eyes are bee strings/bug bites or an allergic reaction (of which you generally never know the cause). Trauma or something stuck in the eye could also cause swelling in the area. Carefully check the area for any stingers or marks. You can apply a cold (not cold) compress to the area. Unfortunately, unless you find a stinger, you won't know the cause. You must also protect the area from rubbing with an elizabethan collar since rubbing could make it worse.
I recommend to play it safe and have Alpha examined by the veterinarian today. The doctor will check the eye as well as look for clues as to the cause. If an allergic reaction is suspected, he will likely receive an injectable antihistamine and possible a steroid or anti-inflammatory. Just like with humans, some pets may experience an anaphylactic reaction. Monitor your dog for signs of an allergic reaction, such as difficulty breathing, general weakness or a progression of swelling. If you see any of those signs this could indicate a serious anaphylactic reaction and you need to take him into the vet immediately. Also, if the swelling doesn't go down after 24 hours, take him into your vet. Best of luck.