The foundation of this decision really mirrors the one we have to make when writing a scientific article; the overarching question is: does the visual do more than words given the same amount of area on the paper? If it does, it's usually a good sign to at least try inserting the work into the text and then perhaps circulate the piece among peers as a litmus test.
As a member in a hiring committee, I seldom see visuals, but we did hire a person two years ago who used a bar graph to indicate her course evaluation scores, which was quite impressive. Looking back, I think I'd probably lay out my evaluation as follows:
Does the candidate make that visual? Among many possible visuals, I'd probably prefer those made by (or designed/inspired by) the candidate. Say, if the author has developed a new health promotion theory that was published, it'd be great to see that expressed in a graph (which is common in health promotion theory.) On the contrary, if the candidate has been working on designing promotion based on a pre-existing theory proposed by someone else, then I'd perhaps not recommend plugging that graph into the research strategy (in that case I would think a citation or mentioning of the original author's name would do the work just fine), unless the said concept is extremely new or novel.
Does the context call for a visual? Some research interests are heavily related to data visualization. For example, if the candidate has extensive experience in applying geographic information system (GIS) to solve research questions, then perhaps attaching a map that published by the candidate is a reasonable demonstration. Another example is field work photos. If you have extensive field research experience, it may also be nice to plug in a representative picture. (Be very explicit about having photo release or permission in the image caption, especially if your picture shows faces of your study participants or minors.)
Is the visual well designed? Now this is what I consider a much more severe risk than going over page limit. By attaching a visual you're also presenting another array of techniques and experiences for evaluation. Yet, criticisms targeting at visuals are, in my opinion, a lot more varying (and something conflicting), compared to those received by written words; this is probably due to acceptably clear grammatical rules for words do exist, but not for visuals. This is not to discourage you from adding a visual even the situation calls for, but to remind all of us to be cautious about the design and adherence to the principles of graphical excellence.
Volumes and volumes were written on graphical excellence, but I would suggest at the very least thumb through William Cleveland's Visualizing Data and Edward Tufte's The Visual Display of Quantitative Information and try to pick and adopt whatever applicable to your work.
More obscure things to pay attention to include: i) Does the image have enough resolution to be printed out nicely? ii) Is the resolution of the image so high that the document's size has become gigantic (e.g. more than 10 Mb)? iii) Would the image be eligible if the committee member prints out the document in black and white?
Closing remarks I don't think I can give a definite yes or no. Adding visuals can be a high risk, high return decision and the chance of success is highly contextual. As someone who is passionate about data visualization, I'd perhaps suggest to stay true to yourself and feel free to experiment a little. If you feel that the visuals really represent your idea and work much better than words, by all means plug in one or two; break the rule a little bit while having some fun expressing yourself. Solicit feedback from peers and advisers to make sure they generally get your ideas.