PoE and Adapter Power Supply Design
I am designing a power supply cicruit. The circuit will get it power from PoE (preferably) and external adapter (7 - 12V).


The first thing that strikes me about your design is that there is no isolation in it. PoE supplies need to be isolated (unless no part of your circuit is exposed to the outside world, but your circuit has a DC input jack, so it clearly has at least one part exposed to the outside world). You probablly want to replace your buck converter with some form of isolated DC-DC.


  because diode will block one of the pair and allow only DC+ and DC- from single pair to supply the current. Am I correct?


Assuming both positive and negative supplies come from the same origial supply, the resistance in both paths is roughly equal and the diodes have roughtly the same characteristics (avoid mixing different types of diode) the positive current will be shared in a roughly equal manner between the two pairs for each rail.


  I am planning to support 7-12V Adapter. I have placed diodes so that single power supply powers my circuit at a single time. To make PoE preferable, the DC-DC converter for PoE outputs 13V and the external adaptor range would be 7-12V. Is my design correct?


Be aware that some adapters labelled as 12V may supply significantly more than that under light load. This mostly applies to older adapters that use a transformer-rectifier design running at mains frequency. If such an adapter is used with your circuit it may end up supplying some or all of the power instead of the PoE.

Though reflecting a commenter I really wonder why you would care which power source supplies the device when both are connected, the diodes will stop the two power supplies from back-feeding each other, who cares if they sometimes end up sharing the load. 


  worried with power dissipation in the diode (and rectifier diodes as well).


Just pay attention to the current ratings of the diodes and if the circuit will be running in a hot environment check the datasheets for any thermal de-rating requirements.

Unlike an AC rectifier the rectifiers in a PoE system will be running with DC input (the diodes are there so PoE still works if pairs are swapped), so make sure you use the continuous DC current ratings when selecing your diodes. 


  I want to use this magjack as I have few samples with me. I am not sure if it work with PoE as nothing has been mentioned in the datasheet.


To be suitible for PoE a magjack needs two special characteristics.


The transformers must be designed to tolerate the DC currents without saturating the core. 
There must be suitable arrangements for the PoE connections, either a seperate PoE connection for each pair to connect to an external rectifier or an internal rectifier.


In general if a magjack is suitable for PoE the manufacturer will explicitly state that. I would interpret the lack of such a statement as evidence that it is unsuitable.

Your jack is definately unsuitable, since it doesn't have the required PoE connections to the cable-side center taps.