Yes, because it is a British Overseas Territory.
the union jack
union jack
I'm not sure what you mean by 'special name', but the official name for the British flag is: Union Jack
The Union Flag of Great Britain.
The Union Flag, it is only correctly termed the 'Union Jack' if it is raised on board a ship.
Flag of Saint Helena was created in 1874.
the Union Jack
The State Flag of Alabama is pre-dated by the "Saint Patrick Flag", the national flag of Ireland, by at least 40 years. It is the flag of Ireland that was incorporated into the Union Flag when Ireland joined the United Kingdom in 1801.
"State Sovereignty, National Union"
Any country's National Flag with a small Union Flag on the upper left corner was once a member of the British Empire.
Around 50 countries used to be part of the British Empire. When they gained independence, some chose to incorporate the Union Flag into their own new national flag.
Strictly speaking, the Jack is only flown on naval ships but people sometimes erroneously referred to the flag as the Union Jack. In recent years, more and more people now refer to it as the Union Flag.