I'd like to edit a page's html source to remove boxes around images with links.
In CSS you can simply assign
text-decoration:none to the element.
I'd like that flexibilty within the SharePoint HTML Source Editor
The short answer: When working with rich text editors that allow you to modify the HTML source, yes.
In other cases you'll find that you get a message saying that SharePoint has modified the source that you explicitly entered and end up removing some of your changes. Sometimes this is due to malformed HTML/CSS code but sometimes it can just be SharePoint trying to ruin your day. The way I have worked around this is by embedding the source another way. Options you have include a Page Viewer Web Part, an iframe, or a Content Editor Web Part with linked content where the linked content could be an HTML or CSS file if you would like.
Needless to say what you are looking to do looks pretty trivial to me and you shouldn't have much of an issue trying to update the CSS inside rich text editor elements that allow for HTML modification, but should you run into issues hopefully what I've outlined above helps you out.
Check the element where the borders are applied, you have to apply your style at that selector. Please add screenshots taking from the dev tool bars for the borders.
The easiest way to accomplish this is to edit the HTML source, find the offending img elements, and add a style attribute with "border:0px;"
The inline style will trump the CSS that adds the borders.