An LED is a very simple device. It behaves according to:
$$I_{LED}=I_{SAT}\cdot\left(e^\frac{V_{LED}}{n \cdot V_T}-1\right)$$
Or, alternately,
$$V_{LED}=n\cdot V_T\cdot \operatorname{ln}\left(\frac{I_{LED}}{I_{SAT}}+1\right)$$
In the above examples, \$n\$ is the emission coefficient (some number that is 1 or larger, but probably not much larger than 10), \$V_T\$ is the thermal voltage (which is \$\frac{k\cdot T}{q}=26\:\textrm{mV}\$ at room temperatures), and \$I_{SAT}\$ is the saturation current (which is the apparent y-axis intercept on a log scale chart based on the slope of the curve representing the voltage vs current of the LED) and is often quite small -- usually much smaller than \$10^{-9}\:\textrm{A}\$.
Suppose, in your case, that the LED is best modeled by \$n=5\$, \$I_{SAT}=1\times 10^{-11}\:\textrm{A}\$ (\$10\:\textrm{pA}\$) and \$V_T=26\:\textrm{mV}\$. Then you could compute:
$$V_{LED}=5\cdot 26\:\textrm{mV}\cdot \operatorname{ln}\left(\frac{600\:\textrm{mA}}{10\:\textrm{pA}}+1\right)\approx 3.226\:\textrm{V}$$
Now, you do NOT get to simultaneously force both the voltage and the current. You can have a power supply that maintains a fixed voltage and simply "complies" with whatever current is needed (up to the specified compliance limits of the power supply.) Or you can have a power supply that maintains a fixed current and simply "complies" with whatever voltage is needed (up to the specified compliance limits.) The LED itself will respond, either way.
I mentioned some "parameter" values above for a hypothetical LED. But LEDs vary all over the place. So let's say that if you grab out a bunch of LEDs and have special equipment that simply prints out the right values whenever you plug in a different LED. Using it you get the following table for six LEDs from the same manufacturer:
$$\begin{array}{r|lr}
\text{LED} \# & n & I_{SAT}\\
\hline
1 & 5 & 10\:\text{pA} \\
2 & 4.8 & 30\:\text{pA} \\
3 & 4.6 & 15\:\text{pA} \\
4 & 5.7 & 18\:\text{pA} \\
5 & 5.3 & 22\:\text{pA} \\
6 & 4.9 & 27\:\text{pA}
\end{array}$$
Let's say you have a power supply that supplies a fixed voltage of \$3.2\:\textrm{V}\$ and does it perfectly. What will be the currents for each of these different LEDs that you hook up? Well, let's look:
$$\begin{array}{r|r}
\text{LED} \# & I_{LED}\\
\hline
1 & 490\:\text{mA} \\
2 & 4100\:\text{mA} \\
3 & 6250\:\text{mA} \\
4 & 43\:\text{mA} \\
5 & 268\:\text{mA} \\
6 & 2190\:\text{mA}
\end{array}$$
Wow! That's bad. All these supposedly similar LEDs produce huge differences in their current using this exact same voltage power supply. And not a single one of them very close to the assumed \$600\:\text{mA}\$, either. Assuming that the power supply can actually deliver over six amps, you could do some serious damage to the LEDs.
Now let's switch over and use a constant current supply designed to provide a fixed \$600\:\textrm{mA}\$ and see what happens with the LED voltage, instead:
$$\begin{array}{r|r}
\text{LED} \# & V_{LED}\\
\hline
1 & 3.23\:\text{V} \\
2 & 2.96\:\text{V} \\
3 & 2.92\:\text{V} \\
4 & 3.59\:\text{V} \\
5 & 3.31\:\text{V} \\
6 & 3.04\:\text{V}
\end{array}$$
Note here that the range of voltages is much smaller! All you need to do is find a constant current power supply that can handle at least \$5\:\textrm{V}\$ or so and you are fine.
Yes, I provided some "clinkers" in the LEDs above. Your specifications said that the LEDs went from \$3\:\textrm{V}\$ to \$3.4\:\textrm{V}\$ at \$600\:\textrm{mA}\$. But that's also the point. While the specifications tell you that it is statistically unlikely to see LEDs out of that range, the fact is that you will still encounter some that are just outside of it from time to time.
This very small variation in voltage is a big reason why "current limiting" resistors work as well as they do. Since the differences in voltage hug a small range, it's very easy to estimate what voltage remains (within a small error range) for a resistor's voltage drop.
If you have a power supply voltage of \$6\:\textrm{V}\$ (not a constant current source, but now a constant voltage source again), then you can be pretty sure that the resistor needs what remains after the LED drop of about \$3.2\pm 0.2\:\text{V}\$. The remainder voltage is then \$2.8\pm 0.2\:\text{V}\$. So if you compute a resistor that will generate the right current given that remaining voltage drop, then the actual current in practice won't vary that much because the remaining voltage drop for the resistor also doesn't vary that much.
(As a note, you can also see here that if you used a constant voltage power supply of \$4\:\textrm{V}\$, that the remainder voltage of \$0.8\pm 0.2\:\text{V}\$ has a much wider variation, percentage wise. And this means that there would be far less consistency in the LED current as a result of that fact. So here, you find that higher voltages for the constant voltage power supply improve current regulation. But this benefit comes at the expense of added wasted dissipation as useless heat.)
A constant current source is often quite similar to a voltage source with an added variable resistor that can adjust itself to drop just the right amount of voltage to keep the current constant. This is done with transistors and/or ICs. But the effect is that instead of a fixed resistor, some added circuitry allows the power supply to vary the resistor automatically, instead. Otherwise, not so different.
Can LED's receive a voltage above their recommended value if the current is constant?... yes, when the constant current is set above the LED recommended value ... do not confuse constant current supply with safety ... you could set the current to 10A (if your supply would allow) and blow up most LEDs connected to it \$\endgroup\$