Hey there, trevor2clerver9! Welcome to tom's hardware and Happy New Year!!! 😀
[strike]What you are asking about is possible, but you'd have to know a few things before you try that out. Basically external drives are the same as 2.5" (like the drives used in laptops for example) internal drives in an external enclosure. They use SATA connections as well, so you should be able to connect them to your motherboard just as any other internal drive.
However, you should know that taking a drive out of its original enclosure would most probably void its warranty. Also, some manufacturers use proprietary connectors, which would make it impossible for you to connect the drive internally or use it with any other external enclosure, besides the one it came with. Another thing you should take into account, is that if the external drive has hardware encryption, the data will be inaccessible without the HDD's original enclosure.
So if you decide to do that, please do so at your own discretion.[/strike]
edit It seems like I've made a terrible mistake, for which I'm sorry. I thought you meant to use an external drive as an internal one. I don't know how I've missed the flash drive part. If the external drive is always connected or has the same letter, you could even set your default download path of your browser to a folder on your external HDD, or simply copy-paste the data.
Hope that helps.
Boogieman_WD