All of my 送別会s are taking their toll, and when you add to that all the favorite restaurants and bars I just HAVE to hit one last time (before I leave for three months), I've been drinking almost EVERY night.
And that's how I picked up these two phrases.
Hair of the dog.
Hair of the dog.
無理強い is kind of the opposite of an extremely common phrase that you'll hear Japanese people say a lot, to be polite, "無理しないでください." Since 無理 means "the impossible," or "the unreasonable," 無理しないで is "Don't try to do more than you can, or more than is comfortable for you." People will say this to each other at meals ("Don't feel like you need to eat EVERYTHING,") or at work, ("Don't overwork yourself"). It's a really useful phrase to know.
無理強い is something that you do not want to be on the opposite end of. It's when people who don't have the tact to say 無理しないでください, insist that you join in the fun, whether it be drinking, karaoke-ing, or smoking marijuana which, to be fair, IS what all the cool kids are doing. As you'll notice, it can have harder meanings (extortion?), but if you use it in the right context, like being hung over, drunk, or a few kilos overweight 無理強いされた, will translate as "I was pressured into it."
迎え酒, on the other hand, is just plain old "hair of the dog." For those of you who aren't native English speakers, or who don't know this expression, "Hair of the dog" is alcohol that you drink when you are hungover. Drinking a beer the morning after drinking ten beers, is supposed to make you feel better. 迎える is to greet, meet, or welcome, so 迎え酒 is pretty easy to understand. It's the sake that comes to pick you up.
Why "Hair of the dog?" Brett looked this one up, and found out that it comes from an old expression/superstition: "The hair of the dog that bit you," was held to help heal dog bites. If you were bitten by a dog, if you could retrieve some of that dog's hair, and put it in the wound, not only would you heal faster, but it was also supposed to prevent infection or disease, like rabies.
I personally prefer this method of dealing with dog bites.