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As a long time Wildlife Rescue Agent and former Animal Control Officer I can tell you that Garter Snakes CAN go a long time without eating (up to a year or more depending upon how much it ate right before you got it) but it is NOT healthy for them. If you caught the snake from the wild...it will probably starve itself to death before it will eat for you.
Many times, wild snakes will not eat what pet store snakes will. If you have put crickets, etc. in the cage...it probably won't eat them. Two suggestions I can give you are to put a small bowl of water with tiny goldfish or guppies into the cage to see if the snake will eat them...or...put a couple of earthworms or red worms in the cage to see if it will eat those since many times that is what a wild Garter Snake eats.
But if the snake does not eat within the next three weeks or so...there is a definate problem. If it is a wild snake that won't eat, you need to let it go. Problem is that now it is beginning to get cold in MN and the snake should have already been in a hibernation den. If you need to let it go, perhaps you could find someone with a dry basement that wouldn't mind if you let the snake go there. It WILL eat bugs and such and would be safe there for the winter.
Here is a good Care Page for your snake: www.anapsid.org/gartcare.html
Let me know if you need more help with this or ask for me again anytime.