What type of gutter screens are attached to the roof itself?
Without being able to see it, I am speculating, but I used to work for a roofing and gutter company and this sounds very familiar.  Please keep in mind that the company I worked with is in the Carolinas and roofing methods do vary by region.  It sounds like you have asphalt shingles.  There are 2 relevant things to know about asphalt shingles.  (1)Along the edge of you roof where the gutters are, there should be 2 layers of shingles, the second one being called a starter course.  (2)On the bottom of asphalt shingles is a bead of sealant, very much like an adhesive, that is designed to heat up when the sun hits the roof and seal to the adjacent row of shingles.

There are many different types of gutter guards (including screens) on the market.  Some of the most popular guards install by sliding the upper lip between the starter course and first course of shingles.  After a few years of being in the sun, the guards can get bonded to that sealant pretty good.  Also singles tend to get brittle over the years and it is probably good that the gutter cleaning company left before they did any damage.

There is also one other situation that I have seen numerous times.  Occasionally, I have run into situations where the guards were not installed correctly.  Particularly, I have seen cases where the installer lifted up the first course of shingles and actually nailed the guards through the shingles.  Most gutter cleaning outfits would run as soon as they see this.  Not to fault gutter cleaning companies, they are just not outfitted to handle such a situation.

I would suggest looking for a roofing company that also does gutters.  They will tend to be larger firms and are likely to offer free evaluations.  I would gather data on the age and type of your roof, gutters, and guards ahead of time if possible.  The reality of the situation is that cleaning your gutters is going to be more involved than the $75 guy with a truck and a ladder is going to be able to handle.  The good news is that if your roof, gutters and guards are set up correctly, they should only need to be cleaned once every 5-10 years.