Untrue, undistilled gin exists.
The difference lies in what it's made with - distilled gin is created by redistilling white grain spirit with juniper berries, whereas compound gin is made by flavouring neutral grain spirit with juniper berries - no redistillation. Granted, both are technically distilled, but they're given different names. The first is called Distilled Gin, as it's redistilled, whereas Compound Gin is not redistilled.
Your martini was likely made with Compound Gin, whereas most are made with Distilled Dry Gin. The only real difference lies with the flavour - what you could of had was an allergic reaction to some of the flavouring used in the Compound Gin. I've never heard of this before, but I did have a coworker once who was allergic to juniper - not the berries, but the tree - and he could only drink extremely high quality gin as the lower brands tended to have a higher content of juniper in them than the expensive brands.
Failing an allergic reaction to juniper, it just sounds like you were hungover. Any allergic reaction would likely affect you immediately, and not the next day. Try the gin again. Be wary of the other ingredients in your martini - if you're using vermouth, be sure it's the kind which doesn't need to be refrigerated (as some DOES need to be!) - if it does need to be, keep it refrigerated. Vermouth can go sour, and while I've never heard of it causing sickness, it doesn't taste good if it goes bad.