In my genealogy research, I can assume with some certainty that if a white southern family had sons born between 1835 and 1846 there is a pretty good chance that all of them will have served in the Confederate armed forces. However, this is not so true as to families in the north. Are there any statistics regarding what percentage of white men, of appropriate age, served in the Union armed forces?
First, according to the 1860 Census, there were 3,526,195 free men, aged 20-39 in the states that would not secede. 3,475,987 of these men were white.
Second, according to the National Parks Service, 2,672,341 men enlisted in the Union Army, 2,489,836 of which were white.
Around 67.8% of enlisted soliders were between the ages of 20-39. To estimate soldiers' age distribution, I downloaded data from the study "Union Army Recruits in White Regiments in the United States, 1861-1865" by Robert Fogel, Stanley Engerman, Clayne Pope, and Larry Wimmer.
So multiplying the number of white soldiers by the percentage within the target age, we get 2,489,836*.678 = 1,688,108 white soldiers age 20-39.
Divide soldiers aged 20-39 by total male population aged 20-39: 1,688,108/3,475,987 = .486
Of course, this estimate is high, because of population growth from 1861-1865. But it usable as an upper bound: Somewhat less than 48.6% of white males, aged 20-39, enlisted in the Union Army.
PS: I left border states in the population calculation, but didn't add in West Virginia (because it obviously wasn't in the 1860 Census). Feel free to play with the denominator as you see appropriate, by using the Census data below:
I have been doing a study of military service in Hawkins County, TN. That is in East Tennessee, where sentiments were divided about the War. But in that county, a much lower percentage of men served than in the rest of the South. I started with the 1860 census and identified every white male between 14 and 45--these were the men who would have been in the 18 to 45 range at some point between 1861 and 1865. Identifying those who served and those who did not from the Compiled Military Service Records, at total of 53% served. Breaking it down, 38% of military age men were in Confederate units and 15% in Federal ones. Those numbers are somewhat inflated because I have added overage and underage men to my service list as I have found them--but there aren't many of those. That participation rate as a percentage of the white population is 11.65%. As a proportion of the total population it is 10.26%.
I believe East Tennessee was very divided--or at least parts were. Six of the 28 ETN counties for which referendum results are available voted for secession (21%). In Hawkins county, while the referendum was defeated 62% to 38%, 1,238 men were enrolled in Confederate field regiments and another 61 in a home guard unit. Of these, 124 (9.5%) later joined a Federal regiment, but 17 of these were true "galvanized Yankees" who joined a unit destined for service on the western frontier in order to escape a prison camp. While some of the Confederates were drafted or dragooned there isn't evidence that many were--and there is evidence that some of the Federal soldiers were, too. In the end, 621 Hawkins men were in Federal service. Subtracting those who changed sides from the Confederate total, that yields 65% in CSA and 35% in USA.
Even if one allows for conscription, dragooning, and suppression of potential Federal enlistment those numbers are evidence of considerable Confederate allegiance. And there may have been a generational difference in feeling as Todd Groce hypothesizes in "Mountain Rebels." More than 30% of the men who served were not old enough to vote in the referendum. Further, voting against secession is not de facto evidence that a man might not go along with state and community once secession was a fact. Lee opposed secession but went with his state: there is no reason to believe that men in Hawkins could not have had similar feelings.
East Tennessee regiments did have significant desertion but that is not all or even mostly attributable to initial opposition to service. By far most desertions occurred after the debacle at Vicksburg, when two ETN brigades were captured, paroled, and not exchanged until the fall of 1863. The exchange didn't take place until ETN was largely under Federal control and many men didn't report to their units. This is as likely or more likely to be from demoralization and the difficulty of passing through Federal lines as it is from initial opposition. At the same time, while there was an increase in Federal enrollment in the county in the second half of 1863, the numbers were far, far lower than the Confederate numbers prior to 1863.
In weighing the evidence I believe one must also consider the impact of propaganda before, during, and after the War. W. G. Brownlow in particular sold the idea of massive Unionism in ETN and that idea served the needs of the community after the war in terms of attracting northern investment.
All of this with the caveat that my research is concentrating on a single county which may not be typical.