Yes, you can add Flash or JavaScript to areas of your site that are configured to accept code. You'll need to upload any necessary files (.swf, .xml, javascript library, etc.) within [Website Management > Data and Media > File Storage] and preface all calls to these files with /storage/ within your code.
You can add code to pages and posts on your site by inserting the HTML for your applet into the Raw HTML editing mode, or clicking the filmstrip icon in the WYSIWYG editor. You can also add this within various areas of your template (from the Code Injection area). To add flash as your site header, for example, you can add the code to the Site Banner HTML Override code injection point if you have the Unlimited plan. We have more information about the pricing/features of each package on our pricing page:
http://www.squarespace.com/pricing
Note: Creating/amending the code for this is an advanced modification that requires programming knowledge to complete. Most issues involving adding this to your site are related to incorrectly constructed file structure.
Sign up for a live workshop.
