Can noxious weeds such as henbit and chickweed be composted?

Yes, but care must be taken so the compost produced does not replant the weeds.

If I put weeds such as henbit and chickweed in a compost bin, it really needs to be covered with a layer of carbon material such as dead leaves. This will eliminate the seeds and spores from blowing. The compost pile at some point need to get to about 150° to kill off the seeds. Piles can easily reach this tempature if kept damp and turned occasionally.  A thorough composting in a hot system should kill most weed seeds.

Special Note: As a general rule, it would be best to not compost diseased plant materials because of the chance of re-infecting your garden.