In addition, it's not clear whether you were entitled to take on outside projects (through your LLC) as part of your engineering work. If moonlighting was not permitted, that could present an obstacle to collecting benefits. As you can't collect unemployment when you're self-employed, it's possible that your state may consider your income through the LLC as having found another job.
Bear in mind, too, that unemployment benefits are only provided for a limited period of time, not until you find your next job. As eligibility may vary from state to state, it's best to check with your state unemployment department or a local attorney who knows this area so that you can get confirmation on whether you'll ultimately be entitled to benefits and if so, for how long.