Can Syria look upon all Palestinian refugees as its supporters?

It can upon the majority but not all.  They're definitely divided.  Some small sections of Palestinians have  unfortunately turned their backs on Syria which had stood by the  Palestinian cause most loyally for decades.  The Western media and many Salafist clerics have purposely misinterpreted this war, portraying it as a  sectarian conflict to heighten religious tensions.  Consequently some Palestinians  within and outside Syria (particularly in Gaza because of the influence  of Hamas' military wing, Ezzedine Qassam Brigade) have been swept away  by the rhetoric of firebrand Salafist sheikhs, selling their loyalty to  the wrong people.  It's  indeed lamentable that some of them have been influenced by Qatar and Saudi Arabia despite the  fact that the Saudi and Qatari stance,  unlike Syria, has always been neglectful to the Palestinian cause.  Considering that such a  fact does exist on the ground, the Syrian government does require  to be discerning and careful.

More fortunately than in Syria, in Lebanon,  Palestinians have been driven by nationalist rather than sectarian  instincts.  That, alongside Hezbollah's efforts to secure the allegiance  of Palestinians in Lebanon, their refugee camps have so far not meddled  in Lebanese sectarian conflicts.

Finian Cunningham explains the regrettable situation concerning some Palestinian refugees in Syria in his article "Syria's shameful betrayal by Palestinians."  

QUOTE: 
In Syria, the dominant anti-government  militants share the same fundamentalist Wahhabi ideology of the Persian  Gulf monarchies, and they have varying affiliation with Al Qaeda. This ideological spectrum has drawn in  factions of the Hamas movement in Palestine, which has traditional links  with the Muslim Brotherhood.   

Why is this particularly shameful for  Palestinians? Well, of all the Arab countries that have provided  humanitarian aid and solidarity to the Palestinians down through the  decades perhaps none has been as loyal and self-sacrificing in its  fraternal help than the Syrian Arab Republic.  From the Zionist Nakba massacres against  the Palestinians in 1948 until the present day, Syria has received  millions of Palestinian refugees with open arms. The country is believed  to host the biggest exiled Palestinian population in whole the region.  

One of the largest Palestinian communities  in Syria is at the Yarmouk refugee camp near Damascus where some 20,000* live. But as political analyst Christof Lehmann points out, the  term  “refugee camp” is misleading.  Palestinian residents  have always been granted full citizenship and civil rights.**  "Yarmouk  is more of an ordinary suburb of Damascus," notes Lehmann, "but it has a technical status of refugee camp under Syrian and international law."  

Since the outbreak of the Syrian conflict  in 2011, Palestinian factions within the Yarmouk district have allowed  the foreign-backed extremists of Jabhat al Nusra and others to  infiltrate and occupy large swathes.

"Half of Yarmouk has been overrun by  Jabhat al Nusrah and some other Al Qaeda brigades, as well as some small  Muslim Brotherhood brigades," says Lehmann. "Sadly, Hamas has been  playing a role in this through a recruitment campaign among young  Palestinians to join the anti-government forces in Syria."

Anti-government Palestinian factions within Yarmouk  blame the Syrian army for the blockade and they claim it is being used  as a tactic to force submission. This narrative has been zealously amplified by the Western media as a way of discrediting the Assad government. But many observers say that it is the  foreign-backed militants who are holding the Yarmouk residents as  hostages and human shields.  This week a UN aid convoy of food trying  to enter the area was turned back when it came under fire from gunmen  believed to be from the militant side.  Palestinian Labour Minister Ahmad Majdalani*** blamed the foreign-backed insurgents.   It's sad that some sections of Palestinians have been swayed because  Saudi and Qatari influence who, unlike Syria, had always been a picture  of neglect to the Palestinian cause. 
UNQUOTE:

 
*The correct figure is150,000 lived in Yarmouk prior to the war, and now some 18,000 remain.

**Syria didn't grant the refugees citizenship, but that didn't make any  practical difference as they were given full civil rights.

***Ahmed Majdalani is actually former labor minister
who  is attached to the PLO and is now a minister without portfolio.  That  explains his Syrian policy.  Unlike Hamas, PLO is supporting the Syrian  government, though latest news through some sources claim that Hamas has  now broken off from Qatar and wants to rejoin Syria.   However, Syria's trust has been shaken quite understandably and while  many Palestinians at Yarmouk are supporting and also voluntarily  fighting for the Syrian Army, the Syrian government is no longer in a  position to blindly trust all residents of Yarmouk Camp ..  unfortunately.