Cryptocurrency Private keys... insiders perspective wanted

I just have one question I really need answering.

If I generate a private key for example from myetherwallet.com and store all of my ether at that location, if my ether wallet happens to shutdown for whatever reason, will i be able to access my ethereum from another website or hard wallet with this private key?

https://i0.wp.com/coolestguidesontheplanet.com/wp-content/uploads/2011/06/ssh-private-public-keys2.jpg?resize=350%2C200&ssl=1

I think I need answers for this before i invest in say a Trezor or Nano S.

Any advice would be much appreciated.
http://az616578.vo.msecnd.net/files/2016/08/29/6360803493185263241013372644_Thank-You-Free-PNG-Image.png