3 added 3 characters in body
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Mathematica, 71 63 bytes

And@@PalindromeQ/@FoldList[1##&,Join@@Table@@@FactorInteger@#]&

Explanation

FactorInteger@#

Factor the input. (e.g. 8668 -> {{2, 2}, {11, 1}, {197, 1}}; for each list in the output, the first element is the prime factor, and the second one is the power.

Join@@Table@@ ...

For each factor-power pair, duplicate the first element by the second element, and flatten the entire thing. ({{2, 2}, {11, 1}, {197, 1}} -> {{2, 2}, {11}, {197}} -> {2, 2, 11, 197})

FoldList[1##&, ... ]

Iterate through the list, multiplying the elements. ({2, 2, 11, 197} -> {2, 2 * 2, 2 * 2 * 11, 2 * 2 * 11 * 197} -> {2, 4, 44, 8668})

And@@PalindromeQ/@ ...

Check whether all of the resulting numbers are palindromes, and apply the And operator. ({2, 4, 44, 8668} -> {True, True, True, True} -> True)

Mathematica, 71 63 bytes

And@@PalindromeQ/@FoldList[1##&,Join@@Table@@@FactorInteger@#]&

Mathematica, 71 63 bytes

And@@PalindromeQ/@FoldList[1##&,Join@@Table@@@FactorInteger@#]&

Explanation

FactorInteger@#

Factor the input. (e.g. 8668 -> {{2, 2}, {11, 1}, {197, 1}}; for each list in the output, the first element is the prime factor, and the second one is the power.

Join@@Table@@ ...

For each factor-power pair, duplicate the first element by the second element, and flatten the entire thing. ({{2, 2}, {11, 1}, {197, 1}} -> {{2, 2}, {11}, {197}} -> {2, 2, 11, 197})

FoldList[1##&, ... ]

Iterate through the list, multiplying the elements. ({2, 2, 11, 197} -> {2, 2 * 2, 2 * 2 * 11, 2 * 2 * 11 * 197} -> {2, 4, 44, 8668})

And@@PalindromeQ/@ ...

Check whether all of the resulting numbers are palindromes, and apply the And operator. ({2, 4, 44, 8668} -> {True, True, True, True} -> True)

2 added 3 characters in body
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Mathematica, 7171 63 bytes

Switch[#,1,True,_?PalindromeQ,#0[#And@@PalindromeQ/Max[#&@@@FactorInteger@#]],_@FoldList[1##&,False]&Join@@Table@@@FactorInteger@#]&

Mathematica, 71 bytes

Switch[#,1,True,_?PalindromeQ,#0[#/Max[#&@@@FactorInteger@#]],_,False]&

Mathematica, 71 63 bytes

And@@PalindromeQ/@FoldList[1##&,Join@@Table@@@FactorInteger@#]&
1
source | link

Mathematica, 71 bytes

Switch[#,1,True,_?PalindromeQ,#0[#/Max[#&@@@FactorInteger@#]],_,False]&