There are several reasons why squares and cubes occur more often than powers of 11, 7, 17, or 4. They are: reduction, apparent three dimensional space of our world, and "simple" relations in our models (laws) of the universe.
We tend to reduce equations into their simplest form. We prefer to cancel out powers and reduce fractions whenever possible, to make it easier to calculate and remember. Even though there are infinite powers divisible by any number, 2's and 3's divide into other numbers more easily than other numbers. Go ahead a pick a number. It will likely be more often divisible by 2 or 3 rather than 17 or 11 or 7. Even if you choose random numerators and denominators in a fraction, and attempted to reduce them, you will likely see more factors of 2 or 3 than these other primes.
Apparent Three Dimensional Space
We appear to live with three dimensions of space. Obviously, any laws which depend on position need to have three variables, and laws the depend on an area need to have two. Our math system tends to work well with cartesian coordinates, which allows us to compare dimensions on equal footing, allowing us to square and cube our dimensions. I suspect (but have not attempted) to survey all known physical laws in their spherical forms, which may reduce the apparent number of 2's and 3's.
Three dimensional space also gives rise to inverse-square laws, such as those which govern the force of gravity or force from an electric field on an object. If we lived in a universe with more or less spacial dimensions, we'd see different powers in these equations.
Most of the models we use to describe the universe have found simple relations between things. This was simple practicality on the part of physicists, but also a lucky break. Sure, you can force students and thought-experiments to proceed by giving them basic information (like a table of position-time pairs), but you could give them slightly more advance information (like velocity) and simplify their lives as well as yours.
Beyond this explanation... I can only say it just so happens to be this way.