Well, after I commented that this might be hard to find, I took 10 seconds and found what looks like a reasonable recent meta analysis, though it's not limited to the US:
Franklin, J. C., Ribeiro, J. D., Fox, K. R., Bentley, K. H., Kleiman, E. M., Huang, X., ... & Nock, M. K. (2017). Risk factors for suicidal thoughts and behaviors: a meta-analysis of 50 years of research. Psychological bulletin, 143(2), 187.
I suspect for most factors the US will follow these data; the specific issue of gun ownership is not included in the paper, though. The paper is quite long and extensive, but I'll pull a couple just for reference (the paper includes ideation, attempt, and death; I'll just report some attempt numbers)...
The top 5 categories were: prior self injury (wOR 4.15), prior suicide attempt (wOR 3.41), positive screening instrument (wOR 2.51), Axis II diagnosis (wOR 2.35), prior psychiatric hospitalization (wOR 2.32).
For some other categories mentioned by the CDC link you gave, family history had an OR around 1.5, physical illness an OR around 2.
There are a lot of caveats to unpack in this meta analysis, and I haven't read through it all, just glanced through, but I think it would be a good read. It has almost 1000 citations according to Google Scholar which is quite a lot for a 2017 paper. Overall, the authors seemed to find the risk factors somewhat disappointing from a prediction standpoint (disappointing in that none are strong enough to identify those at highest risk to intervene).
There are lots of individual papers on suicide risk with gun ownership; most seem targeted at specific populations, such as older people. This one is based on US army soldiers and is specifically for death by suicide:
Dempsey, C. L., Benedek, D. M., Zuromski, K. L., Riggs-Donovan, C., Ng, T. H. H., Nock, M. K., ... & Ursano, R. J. (2019). Association of firearm ownership, use, accessibility, and storage practices with suicide risk among US Army soldiers. JAMA network open, 2(6), e195383-e195383.
Again, they present results from several different categories of firearm access but I'll just grab a couple summary numbers (these are quotes from the abstract):
suicide decedents were more likely to own 1 or more handguns compared with propensity-matched controls (odds ratio [OR], 1.9; 95% CI, 1.0-3.7; χ21 = 4.2; false discovery rate [FDR] P = .08)
Storing a loaded gun with ammunition at home or publicly carrying a gun when not on duty was associated with a 4-fold increase in the odds of suicide death (OR, 3.9; 95% CI, 1.9-7.9; χ21 = 14.1; FDR P = .002)