Is there a way to specify a default option for merging a pull request in GitHub?

You can effectively make this the default for the repository in github by disallowing other types of merges. From the repository, click Settings, and then on the Options tab scroll down to "Merge Button". You can uncheck "Allow Merge Commits" and "Allow rebase merging", which leaves "Allow squash merging" as the only choice for merging pull requests in this repo.

The GitHub user interface will default the option to the last option selected by the user for the current project. This solves the issue of not accidentally using the wrong default, since after the merge is done correctly for the first PR, subsequent merges will have the desired default.

This solution isn't perfect. If a user makes an intentional one-off merge of a different type, they will need to remember to set it back for their next merge, as the default will have changed to what was intended to be a one-off deviation from the norm.