First, I want to say that this is a classic Survival Analysis situation. It's called by different names in different disciplines; while SA is most common, 'Time to Event Analysis' may be more intuitive in your situation. SA can be done in SPSS, although it's been a while since I've had access to SPSS, so I can't advise on the mechanics of performing the analysis, unfortunately. Googling SA SPSS yielded this pdf, which looks like it may be helpful in that regard. I would recommend you use a Cox Proportional Hazard model for this analysis, as you are primarily interested in the effect of your covariates, rather than the baseline hazard rate. These things having been said, I should acknowledge that if a transformation (such as taking the log) sufficiently normalizes your response variable, multiple linear regression should be acceptable. If you are totally unfamiliar with SA, are unlikely to have to work with similar situations in the future, and/or are not in a position to pick up new techniques right now, the costs of trying to learn about SA would outweigh the benefits, and multiple regression would probably be your best bet.
Like others here, I'm generally against categorizing variables. (If you just wanted to know about this as background information, you could read this question and answer where I discuss related issues.) However, running a SA or a multiple regression with an appropriate transform will take care of your issue with your response variable, and I have no qualms with grouping your covariates in this case, given your substantive question. I would create dummies for weekend (i.e., 0 for weekdays & 1 for weekend days), and shift (0 would be the default day shift against which you want to compare, and then one new column / variable for 2nd shift & another new column / variable for 3rd shift). I might also add interaction terms (2 new columns as the products of weekend multiplied by evening shift and by night shift) as I suspect this may be of interest for you. (Specifically, the question might be, 'if nights & weekends are slower, would, say, Sunday night be even slower than I might otherwise suspect given that it's both a weekend and an overnight'?) You could always keep all your days coded, but it doesn't sound like you are intrinsically interested in Tuesday vs. Wednesday per se, just weekday vs. weekend, and this route should give you more statistical power with which to detect an effect that was actually there.
As to your original question regarding why something might be significant when more predictor variables are entered into the model, this has to do with how significance testing works. In multiple regression, you are comparing the slope you found to a null distribution that is estimated using the residual variability. As you enter more covariates into the model, the residual variability decreases. (The residual degrees of freedom also decreases, so it's possible that you could loose power, but more likely you will gain power.)
I would recommend you fit a model with all of the above dummies, and then interpret the results rather than focusing on whether individual covariates are 'significant'. I would also typically advise against running lots of univariate models. Good luck with your project.