This one is a little tricky to answer. According to Varina Davis, in her biography of her husband, she claimed that Jefferson Davis at Lincoln's militia unit's muster, administered the oath of allegiance to Abraham Lincoln. So according to her, the answer is True. Yet according to others, Davis and a couple other officers could not have been dispatched from Ft. Snelling to Lincoln's militia unit's location because they were elsewhere and it was Zachary Taylor who administered the oath. Other online sources also label the meeting between Lincoln and Davis as false.
So, my answer is
False.
Source listed at end:
"In the memoir of Jefferson Davis by his wife,2 it is stated that when this volunteer force
was called out by Governor Reynolds, Gen. Winfield Scott was in command at Fort
Shelling, and dispatched thence to the seat of war two lieutenants to muster in the Illinois
volunteers. One of these lieutenants was said to be a 'very fascinating young man, of easy
manners and affable disposition;' while 'the other was equally pleasant and extremely
modest;' it is further stated that 'a tall, homely young man, dressed in a suit of blue jeans,'
presented himself to the lieutenants as the captain of a company of volunteers, and was
with the others duly sworn in; and that the oath of allegiance was administered to the
'young man in blue jeans' by the 'fascinating' young lieutenant, first named.
This 'fascinating' young officer was Jefferson Davis. who was nearly a year the senior of
Lincoln; his 'extremely modest' colleague was Robert Anderson, who at the beginning of
the War of Secession was in command at Fort Sumter; and the tall, homely, young captain
in 'blue jeans,' was Abraham Lincoln. There may be a grain of truth in this romantic
statement, but it is doubtful. At the time Lincoln was elected captain, and mustered into
service, Scott was not at Fort Shelling; he was in the East, and did not reach Chicago until
July 8.3 Lieut. Jefferson Davis did not, at that time, come from Fort Snelling; he had for
a considerable time been with Col. Zachary Taylor at Fort Crawford (Prairie du Chien).
Neither did Anderson come from Fort Shelling, but from Jefferson Barracks, at St. Louis. It
is possible that Lieutenant Davis administered the oath of allegiance, but I am not 124 aware of any record of such an event. Indeed it is stated upon what is believed to be good authority, that Lincoln and his company were mustered into service by Colonel Taylor himself.1"
Source -
Library of Congress -Abraham Lincoln in the Black Hawk War, pp. 6-7 (footnotes at website)