Can a group health plan deny me coverage based on my current health status?

Sure! They can do anything they want. They want to make money, not help people, really. If you are too sick, you won't do that. However, if they don't cover some illness, like mental illness, but would cover others, then there may be a question of failure to provide "health care parity", which is supposedly illegal. Talk to your agent and find out why. Check with the state department of insurance.