The biggest square

Haskell, 113 121 118 117 bytes

x!s=[0..length x-s]
t#d=take t.drop d
f x=last$1:[s*s|s<-min(x!0)$x!!0!0,i<-x!!0!s,j<-x!s,all(>'0')$s#i=<<(s#j)x,s>0]

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_{-3 bytes thanks to Laikoni!}

_{-1 byte thanks to Lynn!}

_{+8 bytes for the ridiculous requirement of returning 1 for no all-1s sub-matrix..}

Explanation/Ungolfed

The following helper function just creates offsets for x allowing to decrement them by s:

x!s=[0..length x-s]

x#y will drop y elements from a list and then take x:

t#d=take t.drop d

The function f loops over all possible sizes for sub-matrices in order, generates each sub-matrix of the corresponding size, tests whether it contains only '1's and stores the size. Thus the solution will be the last entry in the list:

--          v prepend a 1 for no all-1s submatrices
f x= last $ 1 : [ s*s
                -- all possible sizes are given by the minimum side-length
                | s <- min(x!0)$x!!0!0
                -- the horizontal offsets are [0..length(x!!0) - s]
                , i <- x!!0!s
                -- the vertical offsets are [0..length x - s]
                , j <- x!s
                -- test whether all are '1's
                , all(>'0') $
                -- from each row: drop first i elements and take s (concatenates them to a single string)
                              s#i =<<
                -- drop the first j rows and take s from the remaining
                                      (s#j) x
                -- exclude size 0...........................................
                , s>0
                ]

APL (Dyalog Unicode), 35 34 32 bytes

{⌈/{×⍨⍵×1∊{∧/∊⍵}⌺⍵ ⍵⊢X}¨⍳⌊/⍴X←⍵}

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Adám's SBCS has all of the characters in the code

Explanation coming eventually!

Haskell, 99 97 bytes

b s@((_:_):_)=maximum$sum[length s^2|s==('1'<$s<$s)]:map b[init s,tail s,init<$>s,tail<$>s]
b _=1

Checks if input is a square matrix of just ones with s==('1'<$s<$s), if it is, answer is length^2, else 0. Then recursively chops first/last column/row and takes the maximum value it finds anywhere.

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