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doubt

Source: State Bank of India Probationary Officer Question

I can place A at 14th Jan or 14th March. After that I can't proceed.

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Two possible solutions (not sure if there are more):

A: Jan 14
C: Jan 21
B: Mar 14
D: Mar 21
F: Apr 14
H: Apr 21
E and G: Jun 14 and 21 (both are possible)

Explanation:

As you mentioned, A = Jan 14 or Mar 14. We have similar conditions for D and H; D = Jan 21 or Mar 21 and H = Apr 21 or Jun 21. Also, F's birthday is on the 14th, which implies there's an even number of persons between F and H, and therefore also between C and B. B's birthday is not in January, but it cannot be in June because it's before H but not in the same month. March and April are the only options for B.

We also have that C < F, B; D < G; F < E and B < H.

Now I just started trying, and the first 'guess' was already a bingo:
Suppose B's birthday is Mar 14. Then A's is on Jan 14, C's birthday must still be before B, so Jan 21, and D's is on Mar 21.
Because C and B are adjacent, so must F and H be (and F < H). Because F < E, F cannot be in June anymore so Apr 14 is the only option, which makes H on Apr 21, and E and G are the remaining ones; there is not enough information to determine which one is Jun 14 and which one Jun 21.

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I don't believe there's enough information to come to a single solution. In addition to Glorfindel's two, I've come up with the following, including a complete one.

Jan 14 - A
Jan 21 - D
Mar 14 - C
Mar 21 - G
Apr 14 - F
Apr 21 - B
Jun 14 - E
Jun 21 - H

Based on the following logic:

First, assume that 'H' has a birthday on June 21.
Next, we place 'B' such that there are the same number of dates between 'B' and 'H', as between 'C' and 'F'.
'B' cannot have a birthday on June 14, so we look at the April dates. 'B' cannot be on April 14, as that would leave January 14 for 'C' and March 21 for 'F', which contradicts the rule that 'F' is on an even day. That leaves April 21 as the only option for 'B'.
April 14 is then the only option available for 'F', as 'A' is either January or March 14.
That means that 'C' has a birthday on March 14.
Next, 'A' only has January 14 remaining as a possible birthday.
Also, June 14 is the only possible date for 'E', being after 'F'.
That leaves January and March 21 open. Since 'D' has a birthday before 'G', January 21 is the only possibility for 'D', and March 21 is left for 'G'.

Also possible is the following:

Jan 14 - C
Jan 21 - D
Mar 14 - A
Mar 21 - B
Apr 14 - F
Apr 21 - G/E
Jun 14 - G/E
Jun 21 - H

Starting with the same initial assumption as above,

that 'H' has a birthday on June 21.
March 14 is not possible for 'B' as that would leave January 14 for 'A', and June 14 for 'F', and nowhere to place 'E'.
The only other possible birthday for 'B' is March 21, due to needing the same number of dates between the birthdays of 'B' and 'H' as 'C' and 'F'.
That leaves April 14 for 'F', and thus January 14 for 'C'.
Also, 'D' is left with January 21, and 'A' is March 14.
That leaves April 21 and June 14 for 'E' and 'G'.

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