First off, congrats! I'm still closeted for several reasons but hope to come out around the end of the year. I graduated last May and moved across the country, so I have some idea of what you're going through. its a good idea to be prepared for them to cut you off, but i would hope it wouldn't come to that just because you open up to them about who you really are. You're still their son/daughter who they raised for 21 (or however many) years.
A couple of things: make sure you have a decent bulk of money set up, as moving can be expensive. Not just physically moving yourself/ your things, but at your new place you're going to need a security deposit and first months rent, along with probably some furniture and stuff like that. It adds up quick.
As far as taxes go: did your parents already file for you for this year? If not and you come out in a few weeks and they decide to cut you off, then you may need to handle that yourself. Assuming that you've been a student and made little to no income, and didnt sell any stocks, this should be easy for you and potentially free (at least for your federal return).
For insurance: getting it through your company is great, and as far as your concerned you don't need to do anything special. But your parents are likely paying a little extra to have you on their plan, so they'll want to take you off of it when their plan allows them to make changes. That's on them though, not you.
One thing you might not have considered: your cell phone. You're likely on your parents plan now, so if they cut you off, you'll have to find one on your own. Getting a single plan can be expensive, so prepare at least $40/month for that depending on your needs.
Finally, you may want to consider getting your own credit card. Even with limited credit history, you can often get a card from Capitol one with a low limit and a 0% apr period. This can be useful for moving as you can put some small expenses on it, and you won't ned to pay until after you start getting a salary. Just make sure you read up and use them responsibly, as otherwise you can dig yourself into quite a hole.
All in all, good luck with everything!