No, I doubt anything is wrong with her. From the way she appeared on your video, she's both happy and healthy.
[Speaking as someone who lives in a house with 30+ Eeveelutions, she's got a given amount of...expertise in what a happy one looks like.]
The function you used to identify her as an Eevee, your PokeDex — take a minute to call it up again on your Gear, and this time when you point and click it at her, listen to the extended species information. Doing that with unfamiliar Pokemon will tell you a useful fact about them, along with just their species name.
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[Speaking as someone who lives in a house with 30+ Eeveelutions, she's got a given amount of...expertise in what a happy one looks like.]
The function you used to identify her as an Eevee, your PokeDex — take a minute to call it up again on your Gear, and this time when you point and click it at her, listen to the extended species information. Doing that with unfamiliar Pokemon will tell you a useful fact about them, along with just their species name.