Can my husband and I both claim our two children on our return?

Yes if you are filing a Joint return.  Or how old are they?  If they are over 18 are they full time students?  Do they have any income?  Or are you filing separate returns?

They are 6 and 1. We are planning on filing jointly.

Yes you can claim all your dependents on a Joint return.  A Joint return is one return for the both of you combined.  Is there a reason you are asking?  Do the children have different parents?  Add them under My Info on the left side.  Won't it let you claim them?

A joint return is one tax return that lists your combined incomes, deductions, credits and dependents.  Yes, you list your child dependents on your joint return.  If these are step-children of one of you, you can still list them since step parents have the same ability to claim a child dependent as a biological parent.  

Children are your dependents if they live with you more than half the year and meet a series of other tests that Turbotax will ask about.

The only reason you might not is if there is shared custody with another parent (someone's "ex").  The interview will ask about this too.

Yes, but not exactly. You are not “each” claiming both children. A child may only be claimed once on one return. Since you are filing a joint return, you are “both” claiming the children. But, each child is only being claimed once and  on just one tax return.