Honest opinions 
The flow of the sounds together is nice, but not too keen on either name personally.
For what it’s worth, the two names read really opposite to me. [name_m]Damien[/name_m], while pretty, always struck me as the name of a devilish boy, though I’m not sure why. And [name_m]Eugene[/name_m] is the classic dork name in American culture.
I like [name_m]Damien[/name_m], not particularly a fan of [name_m]Eugene[/name_m], but it has a good flow together.
I actually think they don’t flow that well…for me it sounds awkward, though I love the name [name_m]Damien[/name_m] 
Sorry, but my impression is ‘nerdy demon child.’ [name_m]Dominic[/name_m] [name_u]Jean[/name_u]?
[name_u]Love[/name_u] the name [name_m]Damien[/name_m], but not to crazy about [name_m]Eugene[/name_m]
I have to agree that Damien reminds me of the devil.
Eugene does not…but they are fairly different names.
Dorien
Edgar
Julien
Daniel
Caspar
Simeon
Ezra
I like the combo. I like [name_m]Eugene[/name_m] a lot as a first or middle name. I prefer [name_m]Damian[/name_m]/[name_m]Damien[/name_m] spelled with the ‘an’ at the end but it’s a nice name either way.
[name_m]Damien[/name_m] is the devil’s child in a movie. [name_m]Eugene[/name_m] is nerdy and dated. So it’s a no from me.
I love [name_m]Damien[/name_m]. Before deciding to go with a family name for this baby’s middle name we had our heart set on using [name_m]Damien[/name_m] after St. [name_m]Damien[/name_m] of Molokai. His beautiful story of a life of sacrificial love for the lepers of Molokai is a nice counter to the demon image.
I assume [name_m]Eugene[/name_m] is a family name and not picked for its style factor.
I like [name_m]Damien[/name_m] (it doesn’t remind me of the devil at all) and I think [name_m]Eugene[/name_m] can work. Ultimately, it is a middle name and if some people have a problem with it, it isn’t a big deal. I say this as a person who has a middle name that I don’t really like either.
I think some people are reading too much into it.
I think the names sound OK together - not great, but OK. I think it is the two styles that clash. [name_m]Damian[/name_m] sounds a little edgy and rebellious, whereas [name_m]Eugene[/name_m] is a little nerdy maybe or at least less bold. I think a short middle would sound great.
[name_u]Love[/name_u] [name_m]Eugene[/name_m] :).