Eta is about the proportion of variance explained. If you have an ordinal outcome, you don't have a variance, so I'd say no.
Here's some more explanation. Variance is about how different the scores are. So:
1.1, 1.2, 1.3 has a small difference , hence a large variance.
1, 101, 201 has larger differences, hence larger variance.
1, 2, 10001 has even larger differences, and so the variance is even larger.
But in an ordinal measure, we don't know about differences - all we know about is the order, so for each of those variables, they go in the order 1, 2, 3. The classic example is position in a race - people came first, second, third, all we know about the winner is that they were ahead of whoever came second. Where they 0.1 seconds ahead, or 3 hours ahead. We don't know. So the times could be:
10, 11, 12
Or
10, 100, 101
Or
10, 1000, 1001.
We don't have knowledge of differences, so we don't have variance, so we can't have eta-squared.
You should (possibly) use some form of ordinal logistic regression, then you have options for effect sizes based on likelihood ratios, and/or classification probabilities.
First off, are your two independent variables being adjusted as factors or numerically coded responses and is there an interaction term for the two? The reason I ask is because the test of proportional odds grows very sensitive with small cell counts. For this reason, I often find it justifiable to adjust input variables as their ordinally coded values (1: poor, 2: fair-to-poor, etc.). Doing so allows information to be borrowed across groups, proportionality is assessed so that an associated difference in the odds of a more favorable response comparing units differing by 1 in the predictor are consistent with odds of an even more favorable response (the rough and contrived interpretation of the test of proportional odds).
If your numeric coding still fails to give valid proportionality, it is possible to get consistent cumulative odds ratios estimates by collapsing adjacent categories like the two bottom box responses.
Thirdly, another powered test of association between an ordinal response and two ordinal factors is a plain old linear regression model. Using robust standard errors, you get valid confidence intervals despite the distribution of the errors. This tends to be less powerful that categorical methods, but with fewer pitfalls due to zero cell counts.
Lastly, as a comment, robust standard errors allow consistent estimation of the mean model in most circumstances. I'm not sure if these are implemented in SPSS, but R and SAS use these frequently. As with the proportional hazards assumption in the Cox model, when this "model based assumption check" fails, it does not mean the model results are entirely invalid, it's just that the effect estimates are "averaged" over their inconsistent proportionality. For instance, if proportional odds model has excessive numbers of respondents giving top box responses, and a predictor shows a large association for the top box response but smaller association for other cumulative measures, then you'll find that the cumulative odds ratio is a weighted combination of the several thresholded odds ratios, with a higher weight placed upon the top box OR.
Best Answer
Why do you not compute Spearman's rank correlation coefficient?