Not really. Just start pulling job postings. When I was interviewing with several companies, it really didn't matter what the degree was, as long as it was technically rigorous. Your experience and abilities is what will ultimately land you the job. If I have a degree in EE and another potential candidate has a degree in CS, the one who gets the job is the one that nails the interview questions. You're always going to meet with someone from HR, who isn't really concerned about your technical abilities, but then you're going to meet with your potential manager and peers. They will thoroughly test your technical abilities. If you can impress them, your field of major is immaterial. That said, obviously, a degree in CS will best prepare you for a career in software engineering. Also, it's not as challenging as you might think for someone with an EE degree to move into an SE field. The EE ciriculum at my alma mater has a rigorous programming component.
Below is a list of current job postings on Dice.com for my area. I'm only considering those whose specific title is "software engineer" and posted in the last 30 days.
Software Engineer - Education Required: "BA or BS degree in CS, IT, Computer Engineering , or a related field required"
Software Development Engineer - Education Required: "Minimum BS in Computer Science / Computer Engineering / Software Engineering or equivalent."
Senior Software Engineer - Education Required: "Requires Bachelor's degree in Computer Science or related field and 5+ years related experience. "
Software Engineer - Education Required: "BS in computer science, electrical engineering or related technical field"
Sr. Software Design Engineer - Education Required: "Candidates with CS/EE training or experience strongly preferred."
Software Engineer - Education Required: "MUST have a Computer Science or EE degree. Master's is preferred"
Software Engineer - Education Required: "A Bachelor's Degree or equivalent experience is also required."
Software Engineer - Education Required: "BS in technical field preferred"
The list goes on. I have no reason to lie about this. I've been through it already.
As far as being considered a software engineer, that's what job titles are for. A degree doesn't automatically qualify you as a software engineer.