Wikipedia claims that he refused to shave his beard since it would be "undignified" if he were recaptured (https://en.wikipedia.org/wiki/Maximilian_I_of_Mexico#Downfall). There's a source listed, but a quick Google search doesn't seem to pull up any other mentions of this.
Not only is that reference of Parkes, Henry (1960). A History of Mexico.
But also the following from Hyde, H. - Mexican Empire - The History of Maximilian and Carlota of Mexico
Maximillian agreed to wear spectacles but he refused to cut off his beard, as he said should he be recaptured he would look ridiculous, quoting the example of the Imperialist General Casanova who had removed his moustache...
So it isn't the only reference to it.
You can do additional digging by going to Google Books, http://books.google.com and typing in your search criteria. Often times you will find an excerpt of the source material or other references that can corroborate.
Whether in this case the citations are circular is another matter of course :)
Parkes also notes that Prince Salm-Salm and his wife ( ex-circus performers) bribed the jailers and organized an escape, but Maximillian refused to go unless they accompanied him. This was months after the French had abandoned him; he'd had plenty of time to go, but the Mexican conservatives and his wife thought he should stay ( she even asked the Pope to help) and he listened to them.
You can still get a tour of the convent in Querétaro where he was imprisoned, see his room. It would not have been real hard for him to be smuggled out. Poor guy seems to have been born without a clue.