There are several websites and blogs that purport this 1865 photo at left is Clara Barton and John J. Elwell.  Some even claim "confirmation" of this. 

Considering that Mathew Brady's famous portrait of Clara is also believed to have been taken in 1865, differences of their facial features are obvious--Clara had a much wider, rounder face, with eyes further apart, than that of the woman at left. 

Also, taking into account that Clara never was with or even met John J. Elwell until meeting him at Hilton Head, and they were never in Washington at the same time. Stephen Oates wrote in his book A Woman of Valor, “...he [Gen. Daniel Rucker, Chief QM, Washington, DC] told her that Colonel Elwell had been there inquiring after her, and the possibility that she might see John Elwell again lifted her spirits a little. But she heard nothing more from Elwell until early spring, when she received a letter saying that he was now stationed at the Union prison in Elmira, New York…”
(Comment by Susan Rosenvold, Feb. 24, 2014)

Finally, Clara's personality--her intense shyness which was with her all her life, would not  permitted her to flaunt such a relationship by having a portrait taken of herself with Elwell.  If the purpose was to show a military leader and a nurse of the Civil War, why would they not wear attire to illustrate this?

As for her dress and hair style, this was the style and fashion of tens of thousands of women of this social class in this time period.

At immediate left, a crop of studio portrait of John J. Elwell by Mathew Brady, 1860-1865, from National Archives.

At far left, from Find-A-Grave.

Clearly not the same person in the top photo.

Rather than relying on subjective opinion, judge for yourself by comparing quantitative data.
 

Clara had a slight cleft in her chin, the woman on the left does not.

Clara had moles on her cheeks on each side of her nose, the woman at left does not.

Clara's eyelids droop down at the outer ends, the woman's eyes on the left do not, nor do her eyebrows.

Resizing the photos so that the distance between the pupils are the same (blue lines), then measure the distance from the pupils to the nostrils to the lips, they do not line up.

The green lines are duplicated exactly the same size for both photos.  The vertical distance from Clara's eyes to her nose to her mouth represents a smaller ratio than that of the woman on the left.  Meaning that Clara's eyes are wider apart relative to the distance to her nose and mouth when compared to the woman on the left. 

According to the sources for these photos, both were taken very near 1865 by Mathew Brady.

Finally, adjusting the images to be the same height from hairline to chin shows these are clearly not the same person, considering both were taken in or around 1865.

The woman on the left has a longer face relative to the width.  Clara has a much rounder face and the shape of her eyes is completely different. 

No external proof or documentation is needed, the photos themselves offer the best evidence that this is NOT Clara Barton on the left.

As for the man being John J. Elwell, that too is highly doubtful.  Tens of thousands of men had beards like this for this period of history.  But the evidence that points most to this NOT being J. J. Elwell is the lack of him wearing a military uniform.  It is highly unlikely that a man who served in the Civil War would have posed not wearing his uniform during or shortly thereafter the war, and being a general, John J. Elwell would have proudly worn his uniform with his rank emblems and decorations.

Photos from CivlWarTalk.com, Civil War Couples