Your scale has only 3 levels so it's very unlikely to be conducive to any kind of regular least squares analysis.  You cannot treat it as continuous.

The easiest thing here looks to be a chi-square.  That will tell you if there is a relationship between your measures of any shape.  Unfortunately your measures aren't entirely independent so a regular chi-square test doesn't work and neither does a McNemar test which requires 2x2 tables.  I believe there's a generalization of the McNemar you could use but I'm not familiar with it.  Your best bet for that kind of test is probably bootstrapping the probabilities with a recognition that you have dependence among the measures.

If you're willing to place some restrictions on what you think happens to the relationship you have more options that are easier to calculate.

The suggestion of a logistic regression with the rating as predictor and intention to stay as a response isn't bad but depends on a linear/logistic relationship between the variables such that the probability of yes (or no) increases with rating.  Alternatively you might try an ordinal regression with your rating as response and intention to stay as a predictor.  You should be expecting the relationship to be one of moving higher or lower in the ratings and not some other kind of change (like a more concentrated distribution in the middle).  If your data appear to conform to these limitations you should be OK using either analysis.