I currently live outside of the US and I want my small (one person) software company to be in US jurisdiction. I'm not making enough money yet to justify forming an LLC, so I thought about filing for an assumed business name, preferably in Illinois, where I have a mailbox that I use for business related mail. Is it possible?
Yes, you can register a DBA in Illinois even if you don't live in there. Since you don't have a LLC or corporation, you are simply doing business in Illinois as a sole proprietor. Therefore, you won't need to file a foreign qualification/registration with the Secretary of State as is typical of limited liability entities. However, you will need to register with the Illinois Dept. of Revenue to do business there legally: http://www.revenue.state.il.us/businesses/register.htm
As littleadv mentioned, the DBA registration will need to be done on a per county basis. For reference, here are guidelines from Cook County: http://www.cookcountyclerk.com/vitalrecords/busnamereg/pages/default.aspx
DBA is not a company, its an alias. It just means "you can call Mike T as < whatever your DBA is > ". Generally its per county (at least in California), and you're supposed to file a DBA in every county you're having business (which may be problematic for internet-based businesses, counties all around may claim that you do business there).
If you already have a company and you want to "DBA" it, you have to first register it as a foreign entity with a state.
If you want to set up a company, to be a legal entity that is, you'll have to set up an LLC or a corporation. Depending on the state, the fees might be higher or lower. Some states require LLC's/corporations to file tax returns and some don't.
You should chose the state not by where the mailbox is, but by where the tax consequences are the best for you and the fees are the lowest. Then get a mailbox there.
Edit: Apparently the words "yes you can" were missing in my answer, but that's because its trivial. Of course you can. Its just not what you think it is.