In the 5e PHB I didn't find any specific text about what happens to a druid who breaks alignment, so it seems like we're in house-rule territory here.
I believe that it's important to give a player clear warning before assigning a code-of-conduct penalty. If your druid tries to do something completely inappropriate, give them a warning right there: "Your mental connection to the powers of nature is wavering. Your animal companion turns and growls at you -- it's angry that you're acting against nature. As a druid, you need to act in tune with nature or you could lose your druid powers. Are you sure you want to do this?"
If they go ahead with it, you could treat it similarly as you do for a cleric: they lose their spellcasting and wild shape, and their animal companion stops obeying orders. (Perhaps their animal companion actively opposes them when they try to do something inappropriate.) I recommend letting them get their powers back fairly quickly if they decide they're willing to act like a proper druid; punishing people is not fun, so you should only do it just as much as absolutely necessary.
It's not totally clear from your question, but it sounds like you might have a larger problem, which is that your druid player isn't taking your game seriously -- they're using it as time to act out and be ridiculous. The worry is that, even if you get them not to break their druid oath, they might find other ways to annoy you, such as picking fights with friendly NPCs.
As DM, it's your responsibility to make sure the game is fun, and in some cases that might include asking people to leave the game when they're disrupting it. Try to do this as respectfully as possible -- explain that you're trying to run a serious game, and you don't think their play style is a good fit for your table.