This earlier answer by /u/Chaingunfighter should answer your question. Although I wouldn't lump Pierce and Fillmore together so readily. They both publicly supported the Union, and both were critical of some of Lincoln's wartime policies, but the degree was a bit different. Pierce was much more of a vocal Southern sympathizer than Fillmore was.

Pierce wrote a letter to Jefferson Davis (Pierce's Secretary of War who became President of the Confederacy) in January 1860 in which he encouraged Davis to run for President that fall--the election Lincoln won. In the letter, Pierce gave the opinion that the U.S. needed to "overthrow abolitionism at the polls, and repeal unconstitutional and obnoxious laws [specifically the Liberty Laws]." This letter became public midway through the war, and was met by much criticism in the North.

In the aftermath of the attack on Fort Sumter that began the Civil War, Pierce wrote a letter to Martin Van Buren proposing that all the former Presidents meet and use their collective political standing to head off the war and bring the South back into the Union. If Van Buren were amenable to the idea, then Pierce suggested Van Buren write to the rest of the ex-presidents to call for the meeting, since Van Buren was the most senior ex-President. But Van Buren--firmly in the Lincoln camp by then--rejected the proposal, and wrote that if Pierce believed in it so much, he should call for it himself, and if Pierce did and the others accepted first, then Van Buren would consider attending. Pierce did not pursue the matter further.

Chris DeRose writes in The Presidents' War: Six American Presidents and the Civil War That Divided Them that the proposed ex-Presidents meeting "could only succeed in hamstringing Lincoln at a time when he needed maximum public support, which was likely Pierce’s purpose", and which was the reason that Van Buren diplomatically rejected the idea. It wouldn't have worked anyway--between the time Pierce had sent the letter and received Van Buren's response, Virginia had seceded, with ex-President John Tyler being one of the delegates voting for secession.

And though Pierce was publicly against secession and the war, he mostly played the "both sides" card. In a July 4, 1863, speech, he claimed, "Do we not know that the cause of our calamities is the vicious intermeddling of . . . the northern states with the constitutional rights of the southern states? And now, war! War, in its direst shape—war, such as it makes the blood run cold to read of . . . war, on a scale of a million men in arms—war, horrid as that of barbaric ages, rages in several of the states of the Union . . . and casts the lurid shadow of its death . . . into every nook and corner of our vast domain." He was very outspoken in his criticism of Lincoln during the war, and was at least as ready to blame the Republicans and abolitionists for its outbreak than he was the secessionists, if not more so.

Like Pierce, Millard Fillmore was against some of Lincoln's actions as well, but less publicly vocal, and not as sympathetic to the South as Pierce was. He encouraged compromise to prevent the war from happening, but in the immediate aftermath of the attack on Fort Sumter, Fillmore was elected as president of the Unionists of Buffalo, New York, and said in his acceptance speech:

"We have reached a crisis in the history of this country when no man, however humble his rank or limited his influence, has a right to stand neutral. Civil War has been inaugurated, and we must meet it. Our Government calls for aid, and we must give it. Our Constitution is in danger, and we must defend it. It is no time now to inquire by whose fault or folly this state of things has been produced. The ship of state is in the breakers, the muttering thunder and darkened sky indicate the coming storm, and if she sinks we must go down with her. We have a common lot and must meet a common fate. Let every man therefore stand to his post, and like the Roman sentinel at the gate of Pompeii, let posterity, when the storm is over, find our skeleton and armor on the spot where duty required us to stand."

Later on, Fillmore became critical of some of Lincoln's actions during the war, chief among them the Emancipation Proclamation and Lincoln's suspension of habeas corpus. He stated his position in a letter he wrote to a Columbia University professor:

"I feel it to be my duty to put down this rabid abolition party at the north, which in my opinion is not only aiding the rebels to destroy the Union, but is violating the Constitution by its attempts to abolish slavery, and to establish a military despotism over the loyal states."

However, he tried to temper this with, "I go for the Union and Constitutional freedom, and oppose all those who violate it," indicating his continued support of the North, albeit in a bit of a backhanded way.

Pierce was much more incensed by Lincoln's move, and more direct in his criticism, writing that the Constitution had been "deliberately violated and defied by the national executive sworn to maintain it" and that:

"All of this would have been sufficiently replete with a degree of wrong, disgrace, and honor [sic] which admits of no expression. But what will the world say of a proclamation, emanating from the President of the United States, not only in defiance of the fundamental law of the country for the upholding of which he ought to have been willing to pour his own blood, but in defiance of all law human and Divine which invites the black race in six entire states and parts of several others to use and with all the barbaric features . . . slay and devastate without regard to age or sex . . . the homes of the descendants of men whose fathers fought with our fathers the battles of the Revolution, and whose fathers with our fathers formed and adopted the Constitution . . . the women and children brutally violated and slaughtered shall be white women and children."

When Fillmore wrote to Pierce offering support on the subject, Pierce wrote back that "If this war is prosecuted to abolish slavery . . . what possible justification could be argued for it? A war prosecuted for these objects is itself treason."

At the end of the war, when Lincoln was assassinated, both Fillmore and Pierce gave public speeches mourning his death. Fillmore then essentially retired from public life, while Pierce spent the next couple years lobbying for Jefferson Davis's release from prison.

To read up more on the subject, seek out The Presidents' War: Six American Presidents and the Civil War That Divided Them by Chris DeRose which gives an in depth account of the former Presidents' views and actions before and during the war, and the events and issues that surrounded it.