Can a 10-speed mountain bike chain be used on a 10-speed road bike?
Using the "approved" chain is probably better if all else is equal, and it may be worth a slightly increased cost for more peace of mind. If you already have the mountain-specific chain, there's no harm in trying it. And if the mountain-specific chain is far cheaper and you aren't picky about shifting performance, it will probably be adequate (but no guarantees).

The "mountain bike specific construction" of the HG-X chains seems to be as follows (quotes from Performance Bike, High on Bikes, Amazon, and Competitive Cyclist):


"Dyna-Sys chains have 4 different types of outer plates that facilitate shifting up &amp; down on the cassette or between chainrings." 
"HG-X Chain uses a directional design with specifically shaped inner and outer plates to optimise shifting over the chainset and cassette. Outer plate design has been developed to reduce the chance of chain suck. The inner plate has been developed for better contact with gears providing smoother shifting under high load, giving efficient pedalling, increased rigidity and near silent running."
"Open design of inner links increase mud shedding ability."
"Mushrooming of the pins gives unsurpassed side-plate retention, enabling the chain to far exceed the BS/ISO standard for breaking force."
"Chain is optimised for Shimano 10-speed HG-X MTB drivetrains only."
"Advanced pin riveting technology yields same strength as 9-speed Hollow-pin technology for lightweight Mud-shedding, off-road-specific design."
"Since the action of climbing up the ramps and pins from one chainring to another is slightly different from climbing up one cog to the next, Shimano chose to shape the left and right side plates differently to optimize the chain for each specific purpose."


Given the prevelance of the words "optimized performance" in product descriptions, I would conclude that using the "wrong" chain may affect performance, but it wouldn't cause failure. This conclusion is bolstered by this report from a user who switched from HG-X to a non-Shimano chain (the opposite direction of what you're intending):


"Shifting under load up hills with the Connex is slightly less smooth than with Shimano's HGX chain, but it works just fine.  I'd say it's the same as Shimano's non-HGX chains or any other good top end chain.  This is for sure a nit-picky difference." (Cris LaBossiere)


Of course, you'll never know for sure until you try it on your own bike and judge it by your own standards.