Can single soldiers with no dependents (E4), who have a rental lease or own a home under their name, receive BAH?

BAH is not a right, but a way of supporting your spouse, and kids. If you are single and there is room in the billets, then E5 and below are required to live in the billets- you can be a "ghost" and sleep off post while maintaining your barracks room, but ther will be no BH for it.

I was single no dependents when I left for bct. I received Bah the entire time I was gone (apartment under my name)

At risk of sounding bitter why do airmen receive it and soldiers don't.

I'm sure something must of happened where they trust us with a loaded weapon over a simple lease.

This may help answer your question. http://www.defensetravel.dod.mil/site/faqbah.cfm#Q1.
Also an article about this specific question was written by a Fort Hood Assistant IG (in 2011). http://www.forthoodsentinel.com/news/bah-entitlements-know-the-rules-procedures-before-signing-a-lease/article_82282ea4-8995-552b-8663-3a28a2f11e7e.html
One additional thing to keep in mind, check with your chain of command because most commanders have established local policies regarding on/off post housing rules.

A few factors go into this question, if the soldier is National Guard or Reserve they will receive some type of BAH when on orders regardless of dependents. But your question is for Active Duty? when given orders typically you can terminate the lease, owning it is another ball game though.

Normally how it works for active duty as in the regular Army is that service members in the rank of Sergeant and below will not receive BAH if there are rooms available or have it approved by higher.

Yes

Typically not unless the barracks are over occupied and the Army has told you BAH is headed your way. If they give you a barracks room though, as a single soldier w/o dependents, you wont receive BAH.