The conceptual uses of "square" and "squared" are subtly different, although (almost) interchangeable:
"Squared" refers to the past action of taking or computing the second power. E.g., $x^2$ is usually read as "x-squared," not "x-square." (The latter is sometimes encountered but I suspect it results from speakers who are accustomed to clipping their phrases or who just haven't heard the terminal dental in "x-squared.")
"Square" refers to the result of taking the second power. E.g., $x^2$ can be referred to as the "square of x." (The illocution "squared of x" is never used.)
These suggest that a person using a phrase like "mean squared error" is thinking in terms of a computation: take the errors, square them, average those. The phrase "mean square error" has a more conceptual feel to it: average the square errors. The user of this phrase may be thinking in terms of square errors rather than the errors themselves. I believe this shows up especially in theoretical literature where the second form, "square," appears more often (I believe: I haven't systematically checked).
Obviously both are equivalent in function and safely interchangeable in practice. It is interesting, though, that some careful Google queries give substantially different hit counts. Presently,
"mean squared" -square -root -Einstein -Relativity
returns about 367,000 results (notice the necessity of ruling out the phrase "$e=m c^2$" popularly quoted in certain contexts, which demands the use of "squared" instead of "square" when written out), while
"mean square" -squared -root -Einstein -Relativity
(maintaining analogous exclusions for comparability) returns an order of magnitude more, at 3.47 million results. This (weakly) suggests people favor "mean square" over "mean squared," but don't take this too much to heart: "mean squared" is used in official SAS documentation, for instance.