Any thoughts on switching insurance while pregnant?

I'm not really sure how this works! We are moving out of state, and are pregnant. I think I can keep our current coverage and use it out of state.. but only if I don't tell them we're moving, right?
I have been told I may have problems starting new coverage because the unborn baby is an 'unknown entity' or something?!

Any thoughts would be greatly appreciated!

Confused,
Vida