If you were greatly satisfied by the results from Costco, then you will likely be happy with the results from PrintNinja. Both parties have to do color conversions to CMYK and similarly rooted ink systems for images received in RGB (any other color space as well). The difference here is that Costco didn't tell you they were going to do it implicitly.

PrintNinja's website states two options for color conversion from RGB to CMYK, one implicit and one explicit.

Method 1: Let Us Convert Your Colors For You (Less Control)

The simplest method is just to submit your RGB files and let us convert them for you. We’ll show you the CMYK files in your Electronic Proof before going to press, so you’ll be able to see exactly how your colors have changed in the conversion process.

However, this method provides the least amount of control for you, since we won’t make any changes to your colors beyond the conversion. If you want to be able to tweak your colors after conversion, you should convert them on your end before uploading your artwork.

Method 2: Convert Colors Yourself (More Control)

If you want the most possible control over your colors, we recommend you convert colors on your end.

If you do nothing further, they will proceed with Method 1, and you can expect the results I've mentioned above.

If it's not too late to resubmit your book, you can do the following to extract the most fidelity from your printing process, and use PrintNinja's second option:

1. If possible, export your images again from your workflow as 16-bit TIFFs using the "Japan Color 2001 Coated" ICC profile (as mentioned on PrintNinja's page). The ICC profile can be obtained from Adobe here.
2. Have your designer replace the old images in the InDesign file.
3. If you have any colored graphics, such as horizontal rules, icons, or vector art in your book, have your designer check for color values that are not specified in CMYK, and select replacement values in CMYK. PrintNinja recommends a number of specific CMYK values for best results.
4. Also, verify with your designer that the text (presumably in black) in your book is specified using only a K-value (100 or 100% => C=0,M=0,Y=0,K=100), and not the synthetic value (C=100,M=100,Y=100,K=0-100) or any such analogue. Rich Black and similarly constructed color values can cause issues with ink bleed in print.
5. Last, your designer can export a new pdf file that you can submit to PrintNinja.

One last note, if you wanted to do the same with Costco, they recommend following the preparation guide here for in-store printing; and using profiles from here for send-out printing.