I think a lot depends on the specifics of the program.
Clinical psychology programs are quite competitive to get into with typical admissions being on the order of about 5% of the applicants (at least in Canadian programs - I'm assuming it's similar in the US). I think they tend not to require a specific BSc, so you would probably be qualified to apply to a number of programs, but I believe the admissions people tend to look for very specific traits in their candidates including:
- a solid background in psychology
- a background in biology and neuroscience
- some humanities courses that explore the diversity within the human population
- research experience in psychology
- exposure to clinical work (e.g. volunteer experience in peer counselling)
- very high GPA in all of this and high GRE scores
Typically engineering students are not likely to have this kind of background because the course load in engineering is quite demanding and any extra-curricular project-type work they do tends to be 'engineering' oriented. That's not to say that an engineering student couldn't do all of this. If you were a first year student who wanted to do this, it would be tough, but possible to build your education so that you would be competitive even through an engineering program. But in most cases I'm not sure that one could be "retroactively" competitive.
All of that said, there I suspect that non-clinical psych MSc degrees would be a little more flexible in who they would admit. The issue though is that if you want to get into the clinical counselling (i.e. those jobs where there just aren't enough psychologists to go around) they wouldn't be training you for this specifically. You'd get through the program after a couple of years and $x and likely not be any more qualified for professional positions than you are now.